List of top Mathematics Questions on Properties of Determinants asked in JEE Main

If \(\alpha \neq a\), \(\beta \neq b\), \(\gamma \neq c\) and \[ \begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0,\] then \[ \frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c} \] is equal to:

If for some $c < 0$, the quadratic equation, $2cx^{2}-2\left(2c-1\right)x+3c^{2}=0$ has two distinct real roots, $\frac{1}{a}$ and $\frac{1}{b},$ then the value of the determinant. $\begin{vmatrix}1+a&1&1\\ 1&1+b&1\\ 1&1&1+c\end{vmatrix}$ is :